Particles on a periodic box, a probe neighbourhood, live order parameter and the phi(eta) noise sweep against the all-to-all exact curve.
Sweeps eta from 0.2 to 6.0 at N = 300 with your density, speed and radius; then checks the all-to-all exact value sin(eta/2)/(eta/2) and that the local model never exceeds it.
Not run yet.
The science
Model. N point particles move at constant speed v0 on an L×L torus (L = √(N/ρ)). At every step each particle takes the mean heading of all neighbours within radius r (itself included) and adds angular noise: θi(t+1) = arg Σj eiθj + η(u − ½), u uniform on [0,1); then xi += v0(cosθi, sinθi). In vectorial noise the random vector is added to the neighbour sum before taking the angle (Chaté et al. 2008). The ring and links around the probe particle (white) show which neighbours it averages over.
Order parameter. φ = |(1/N) Σ eiθj|. Random headings give the random-walk floor ⟨φ⟩ = √π / (2√N). Low noise gives φ → 1 (a flock), high noise gives the floor; the lower-right graph is the sweep φ(η).
Exact reference. If the interaction radius exceeds the box every particle sees all others, shares one mean angle, and then φ is just the length of the average noise vector: φ = sin(η/2)/(η/2) (2/π = 0.637 at η = π). This all-to-all curve is an upper envelope for the local model, and the experiment tests both the envelope and the exact value.
Transition. Vicsek et al. (1995) saw a continuous order–disorder transition; Grégoire & Chaté (2004) showed it becomes discontinuous (travelling dense bands) in large systems; vectorial noise is first order even in small boxes. This 2D page uses small systems (N = 300 for the sweep) where the crossover looks smooth — so it estimates a crossover noise, not a critical exponent. Use the 3D lab for susceptibility and Binder-cumulant scaling.
Honest simplifications. Synchronous updates, no inertia or repulsion, a minimal model rather than a real fish school or bird flock; sweep values are single time-averages with modest statistics (about ±0.03 in φ).